Multiscale methods for Levitron Problems: Theory and Applications

TL;DR

This paper introduces a novel multiscale splitting method for accurately simulating the Levitron problem, combining higher order schemes and extrapolation techniques to improve stability and precision near equilibrium states.

Contribution

The paper presents a new higher order splitting scheme with multiproduct expansion for multiscale Hamiltonian systems, specifically applied to the Levitron problem.

Findings

Iterative and extrapolated Verlet schemes outperform Runge-Kutta methods.

The proposed methods achieve stable states near the Levitron's equilibrium.

Numerical results demonstrate improved stability and accuracy of the new integrators.

Abstract

In this paper, we describe a multiscale model based on magneto-static traps of neutral atoms or ion traps. The idea is to levitate a magnetic spinning top in the air repelled by a base magnet. For such a problem, we have to deal with different time and spatial scales and we propose a novel splitting method for solving the levitron problem. We focus on the multiscale problem, which we obtain by coupling the kinetic T and the potential U part of our equation. The kinetic and potential parts, can be seen as generators of flows. The main problem is based on the accurate computation of the Hamiltonian equation and we propose a novel higher order splitting scheme to obtain stable states near the relative equilibrium. To improve the splitting scheme we apply a novel method so called MPE (multiproduct expansion method), which include higher order extrapolation schemes. In numerical studies, we discuss the stability near this relative equilibrium with our improved time-integrators. Best results are obtained by iterative and extrapolated Verlet schemes in comparison to higher order explicit Runge-Kutta schemes. Experiments are applied to a magnetic top in an axisymmetric magnetic field (i.e. the Levitron) and we discuss the future applications to quantum computations.